The following is from Fulton's Intersection Theory:
Theorem 6.7 (Blow-up Formula). Let $V$ be a $k$-dimensional subvariety of $Y$, and let $\widetilde{V} \subset \widetilde{Y}$ be the proper transform of $V$ , i.e. the blow-up of $V$ along $V \cap X$ . Then $$f^{*}[V] = [\widetilde{V}] + j_{*} \bigl \{c(E) \cap g^{*}s(V \cap X, V) \bigr \}_{k}$$ in $A_{k}\widetilde{Y}$.
$$ \require{AMScd}\begin{CD} \widetilde{X} @>j>> \widetilde{Y} \\ @VV g V @VVfV \\ X @>i >> Y \end{CD} $$
Where $f : \widetilde{Y} \longrightarrow Y$ is the blow up of $Y$ along $X$ with exceptional divisor $E$.
Now, for $Y = \mathbb{P}^{n}$ and $X \subset Y$ a smooth subvariety the second answer to the question {https://mathoverflow.net/q/72710} we have
1) $\widetilde{H}^{n} = 1$,
2) $\widetilde{H}^{n-i} \cdot E^{i} = 0$ for $i < c = \text{codim}(X)$,
3) $\widetilde{H}^{n-i} \cdot E^{i} = (-1)^{i-1}s_{i-c}H_{X}^{n-i}$ for $i \geq c$,
where $\widetilde{H} = f^{*}H$ is the pull-back of hyperplane class $H$ on $Y$, and $H_{X} = H\cdot X$.
Do these equalities come from the above theorem? How?
Every help is welcome.
Thanks a lot.